I read an interesting essay about the "New Math" effort from several decades ago: Changing School Mathematics by Robert Davis (about 12 pages long). The essay gave an overview of New Math but focused on the Madison Project, a specific effort at implementing New Math, and how it sought to change the way students learned math from the old style of following formulas to a new style of emphasizing conceptual understanding. The essay covered some interesting techniques of teaching and how students responded, and I some of the thoughts from the essay really resonated with me.
The project was set up to address some shortcomings in the popular math teaching of the time and the difficulties that math education faced. One of the biggest problems in teaching mathematics, particularly at the primary education levels, is that it is both difficult to convey good understanding of mathematics and to learn if the student has gotten a good understanding or if they are simply memorizing and repeating steps. Much of the usefulness of math stems from understanding what is happening and why it is happening and a student who doesn't understand that is missing out on the majority of what learning math has to offer. So there has been a lot of debate and experimentation about the "right" way to teach math and what styles are or are not effective.
I liked a lot of what the author had to say about teaching math, and I think a lot of it applies to learning and teaching in general. A lot of it resonated with my inner mathematician (I'm a math major) and reminded me of ways that I like to learn and ways that I think are effective in tutoring other people in math. (Teaching math is a subject I'm very interested in right now, as I think about starting to teach my children math in the next few years.)
Here are some interesting quotes from the article, although I recommend reading the whole thing. I think that a lot of it applies to most learning topics in most learning scenarios, not just math in a class room. (Emphasis in quotes is preserved.)
No child can be expected to "discover" historical accidents or what is in the teacher's mind. Only after a task is clearly understood can the creativity and inventiveness of children take over the agenda. The correct understanding of the [Madison] project's approach might better have been stated as, "If, at this moment in the lesson, what is needed is an intellectual breakthrough of some sort, please wait and let a student take the first step." If you wait, someone will.
- pg. 637
I like this point because I'm very convinced that we both understand and remember things better when we make the connections ourselves. It's certainly possible to feed people answers prematurely, either for the sake of keeping up a pace or because we're too excited ourselves and we want to share our knowledge, and in doing so we hinder the listener's ability to play with key pieces of the puzzle themselves. Providing an answer too soon can deny the listener a "light bulb" moment, which is key for understanding ideas.
When I'm explaining an idea to somebody, perhaps one that itself involves several different ideas, there will inevitably be a time where a few ideas come together and the listener doesn't instantly make sense of everything. (This arises frequently, almost whenever a relatively deep conversation has gone on for 10 or so minutes.) They're not dumb, just taking a moment to sift through things. When I feel like this has happened, I usually like to just go quiet for a moment and let the other person think. Even if they ask a question, I may just ignore it for a little bit (politely, of course, perhaps with a little "hm..." and a pregnant pause). The idea will make more sense to them and be more useful if they put it together in their own head, and I'm usually certain that nothing I can say will substitute for their own effort.
Besides demonstrating the assimilation paradigm, the preceding example shows another way in which the project provided help to students: the use of clear, unambiguous language and notations. We had been aware that David Page was using small raised symbols for positive and negative, as in +2 or -3, and carefully using the words positive and negative when that was the idea (as opposed to plus and minus in the situation where those meanings were intended and nonraised symbols were used). We had not chosen to follow his example in this usage until some seventh graders, asked to invent a sensible way to add, subtract, and multiply signed numbers, responded that
(+2) x (-3)
should be equal to 3. How come? "Because two times three is six, and then you have to subtract three." We converted immediately to Page's notation, and this difficulty disappeared.
Probably, if we were telling the students what to do, alternative interpretations such as the one above might not arise --or, at least, might not see the light of day and might not come to be noticed. But if the children themselves are building up the mathematics, if they are inventing ways to proceed, then exactly how they are thinking about the ideas becomes directly relevant. In traditional "teaching by telling," the question of which notation is used may be seen as unimportant.
- pg. 640
When a teacher is emphasizing a conceptual understanding of math, clear notation is necessary. This example looks like a very interesting case where seemingly simple notation obscured the math problem. The student tried to derive a mathematical meaning from the notation, but the notation was imperfect. It was a simple example, and the author points out that this is easily solved by just telling the student what to do (they will eventually memorize the rules through repeated practice), but it illustrates how notation can be an artificially imposed difficulty.
I think that good notation should distinguish between what something is and what actions are happening. There is probably a more formal way of expressing that thought, but that's my basic idea. In the example above, the student confused what quantity existed (negative 3) for what action needed to be done (subtracting 3). Obviously, the two ideas are closely related, but technically the notation (aka, "-3") was ambiguous by itself, the intent had to be inferred from the context, and the interpretation mattered. (This last point is important because, particularly in higher math, ambiguous notation may be permitted when various interpretations/definitions are equivalent, so the reader is free to choose the interpretation they like best.) Such seemingly simple things can, at a minimum, create confusion, and once the student is confused their progress will be slowed down. I don't really see a need for such ambiguous notation, and in an ideal system we wouldn't have it. Since notation is mostly established by popular tradition, it's no surprise that odd quirks like this exist.
The essay closed with:
By imagining that mathematics means knowing when to invert and multiply, we have come to trivialize mathematics, knowledge itself, and even the nature of human thought. Anyone who listens carefully to what children really think about the world will know otherwise.
Another re-iteratation of the thesis: Students, even young ones, can understand ideas and find motivations for those ideas on their own given sufficient guidance. Rote memorization of using tools doesn't teach the same skills.
Math can be a difficult subject to teach. One reason it's hard is because a good understanding of it is generally a very "internal" feeling. Mere words don't really convey what a person understands about math, much like words fail when trying to describe a piece of music. Differences of viewpoints, preferred approaches, and mental strengths between teachers and students can raise barriers when trying to take knowledge from the teacher's head and get it into the student's head. But it's not an impossible task, and there are definitely some techniques have tend to produce better success rates than others. Good teaching will allow the student to build math understanding in their own head, instead of cramming a pre-defined structure in there. The Madison project tried to accomplish this, and it seems like they had a lot of good ideas that students of any generation would benefit from.